Like my old buddie has said get in touch with higher power or aeroservice, I hear aeroservice got a deal going on for the B737 type. You might need to get the SIC (second in-command) rating like NDB mentioned, so make sure you check that out very well before u move in for the course, NDB should be able to sort u out with details.
Conversion is no big deal, you just come back with all your papers from the FAA and the NCAA will take it from there. Some delays for your exam results and all that but no big deal, it all comes together at the end of the day.
Goodluck and don't delay.
Balewa